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Register a new userHomoclinic orbits and entropy for three-dimensional flows
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We prove that every $C^1$ three-dimensional flow with positive topological entropy can be $C^1$ approximated by flows with homoclinic orbits. This extends a previous result for $C^1$ surface diffeomorphisms cite{g}.
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We construct symbolic dynamics on sets of full measure (w.r.t. an ergodic measure of positive entropy) for $C^{1+epsilon}$ flows on compact smooth three-dimensional manifolds. One consequence is that the geodesic flow on the unit tangent bundle of a compact $C^infty$ surface has at least const $times(e^{hT}/T)$ simple closed orbits of period less than $T$, whenever the topological entropy $h$ is positive -- and without further assumptions on the curvature.
Let ${T^t}$ be a smooth flow with positive speed and positive topological entropy on a compact smooth three dimensional manifold, and let $mu$ be an ergodic measure of maximal entropy. We show that either ${T^t}$ is Bernoulli, or ${T^t}$ is isomorphic to the product of a Bernoulli flow and a rotational flow. Applications are given to Reeb flows.
We study the cohomological pressure introduced by R.Sharp (defined by using topological pressures of certain potentials of Anosov flows). In particular, we get the rigidity in the case that this pressure coincides with the metrical entropy, generalising related rigidity results of A.Katok and P. Foulon.
It is proved that a certain type of monotone flow has a global period provided periodic points are dense.
Let X be a connected open set in n-dimensional Euclidean space, partially ordered by a closed convex cone K with nonempty interior: y > x if and only if y-x is nonzero and in K. Theorem: If F is a monotone local flow in X whose periodic points are dense in X, then F is globally periodic.
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